Theorems · Theorem · group theory

PSL.iwasawaT_map_conj

∀ {F : Type u_1} [inst : Field F] {ι : Type u_2} [inst_1 : DecidableEq ι] [inst_2 : Fintype ι]
  (g : Matrix.SpecialLinearGroup ι F) (H : Subgroup (Matrix.SpecialLinearGroup ι F)),
  Subgroup.map (QuotientGroup.mk' (Subgroup.center (Matrix.SpecialLinearGroup ι F))) (MulAut.conj g • H) =
    MulAut.conj ↑g • Subgroup.map (QuotientGroup.mk' (Subgroup.center (Matrix.SpecialLinearGroup ι F))) H

The SL-level equivariance pushed through the quotient: the image in PSL of the conjugate MulAut.conj g_SL • H equals MulAut.conj (mk g_SL) • (image of H).

Defined in: Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer
Cited by: 0 results in Mathlib
Foundations: Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field DecidableEq Fintype

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DFunLike.coestatement and proof · cited by 62,936
Fintypestatement and proof · cited by 7,736
Fieldstatement and proof · cited by 7,404
mul_oneproof · cited by 3,885
MonoidHomstatement · cited by 3,629
Subgroupstatement and proof · cited by 3,593
HasQuotient.Quotientstatement and proof · cited by 2,301
mul_assocproof · cited by 1,667
Matrix.SpecialLinearGroupstatement and proof · cited by 348
Subgroup.mapstatement and proof · cited by 301
QuotientGroup.mkstatement and proof · cited by 196
MulAutstatement · cited by 158

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